What @clive has mentioned is great if you want to use `/user` for this, but from a number of angles, I really don't like using it at all for this kind of thing.  I would like to say mostly from a style perspective, but actually more from a linking perspective, eg, an email or post that includes `check out my cool profile on mototribe:  www.mototribe.com/user` bollixing things up.

So, when I've done stuff like this, I include php logic in my block view along the lines of:

    $global $user;
    $block=array();
    $node=menu_get_object();

    if ($node->type==THENODETYPEWEARELOOKINGFOR && $node->uid==$user->uid) {
      // generate the block here
    } else {
      // don't generate anything (or put something else in it's place)
    }

    return $block;

so the block is only generated when the currently logged in user looks at a certain node type that s/he owns.  Obviously I've not fleshed out all the isset()s or empty()s needed in prime time for the above but I am sure you get the jist.

Of course, I could also be talking total smack here if D7 user profiles aren't nodes, but this works great in D6 with content profiles :)